Signals and Systems - Electrical Engineering

472 C H A P T E R 8: Discrete-Time Signals and Systems

Discrete-Time Unit-Step and Unit-Sample Signals

The unit-stepu[n]and the unit-sampleδ[n]discrete-time signals are defined as

u[n]=

{

1 n≥ 0

0 n< 0

(8.21)

δ[n]=

{

1 n= 0

0 otherwise

(8.22)

These two signals are related as follows:

δ[n]=u[n]−u[n−1] (8.23)

u[n]=

∑∞

k= 0

δ[n−k]=

∑n

m=−∞

δ[m] (8.24)

It is easy to see the relation between the two signalsu[n] andδ[n]:

δ[n]=u[n]−u[n−1]

u[n]=δ[n]+δ[n−1]+···

=

∑∞

k= 0

δ[n−k]=

∑n

m=−∞

δ[m]

where the last expression is obtained by a change of variable,m=n−k. These two equations should

be contrasted with the ones foru(t)andδ(t). Instead of the derivative relationδ(t)=du(t)/dt, we

have a difference relation, and instead of the integral connection

u(t)=

∫t

−∞

δ(τ)dτ

we now have a summation relation betweenu[n] andδ[n].

RemarksNotice that there is no ambiguity in the definition of u[n]orδ[n]as there is for their continuous-

time counterparts u(t)andδ(t). Moreover, the definitions of these functions do not depend on u(t)orδ(t),

and u[n]andδ[n]are not sampled versions of u(t)andδ(t).

Generic Representation of Discrete-Time Signals

Any discrete-time signalx[n]is represented using unit-sample signals as

x[n]=

∑∞

k=−∞

x[k]δ[n−k] (8.25)